THE KELLER-SEGEL MODEL FOR CHEMOTAXIS WITH PREVENTION OF OVERCROWDING: LINEAR VS. NONLINEAR DIFFUSION

Transcription

1 THE KELLER-SEGEL MODEL FOR CHEMOTAXIS WITH PREVENTION OF OVERCROWDING: LINEAR VS. NONLINEAR DIFFUSION MARTIN BURGER, MARCO DI FRANCESCO, AND YASMIN DOLAK Abstract. The aim of this paper is to iscuss the effects of linear an nonlinear iffusion in the Keller-Segel moel of chemotaxis with volume filling effect. In both cases we first cover the global existence an uniqueness theory of solutions of the Cauchy problem on R. Then, we aress the large time asymptotic behavior. In the linear iffusion case we provie several sufficient conitions such that the iffusion part ominates an yiels ecay to zero of solutions. We also provie an explicit ecay rate towars self similarity. Moreover, we prove that no stationary solutions with positive mass exist. In the nonlinear iffusion case we prove that the asymptotic behaviour is fully etermine by the iffusivity constant in the moel being larger or smaller than the threshol value ε =. Below this value we have existence of non-ecaying solutions an their convergence (in terms of subsequences) to stationary solutions. For ε > all compactly supporte solutions are proven to ecay asymptotically to zero, unlike in the classical moels with linear iffusion, where the asymptotic behaviour epens on the initial mass.. Introuction This paper focuses on the mathematical analysis of a chemotaxis moel in the cases of linear an nonlinear iffusion. The general structure of the moel we consier is ρ = (M(ρ) µ(ρ, S)) t (.) S + S = ρ, pose on R R + subject to the initial conition ρ(x, ) = ρ (x), ρ L (R ), ρ (x), x R. (.) The mobility term M is given by M(ρ) = ρ( ρ). Such a choice takes into account the prevention of overcrowing effect, sometimes also referre to as volume filling effect (see the motivations an the references below). The potential µ reas where δe δρ to ρ. µ(ρ, S) = δe (ρ, S) (.3) δρ enotes the functional erivative of some energy functional with respect

2 M. BURGER, M. DI FRANCESCO, AND Y. DOLAK If we moel the energy as a combination of a logarithmic entropy an an aggregation part, i.e., E(ρ, S) = ε (ρ log ρ + ( ρ) log( ρ)) x R ρs x + ( S + S ) x, (.4) R R with ε >, then system (.) becomes ρ = (ε ρ ρ( ρ) S) t S + S = ρ, (.5) which is a special case of the Keller-Segel moel for chemotaxis, escribing the behaviour of a cell population ρ uner the influence of the chemical S prouce by the cells themselves. Introuce in 97 [KS7] to escribe aggregation of slime mol amoebae, this moel has become one of the most wiely stuie moels in mathematical biology. The cell flux on the right han sie of (.5) comprises two counteracting phenomena: ranom motion of cells escribe by Fick s law an cell movement in irection of the graient of the chemical S. In contrast to the equations presente here, the graient of S is multiplie by a linear instea of a nonlinear function of ρ in the classical version of the moel. An interesting feature of this choice is the fact that solutions can become unboune in finite time, thus giving rise to concentration phenomena. Whether this blow-up of solutions occurs or not epens typically on the initial ata an the space imension, an conitions for the blow-up of solutions have been erive by a large number of authors, see for instance [JL9, HV96, HMV97, GZ98, Vel]. Most stuies focus on moels where the evolution of the chemical S is governe by a parabolic equation (as in the original Keller-Segel moel). We shall often refer to this moel as the fully parabolic case. Typical alternative formulations for the evolution of S are given either by an elliptic equation in a more general form than that in (.) or by the Poisson equation. In the majority of the cases, the moel is consiere on boune omains, typically with Neumann bounary conitions. Concerning the case of an unboune omain, an extensive analysis of the moel on R,, is performe in [CPZ4] (see also the survey [Per4]), where the authors show that the global existence of solutions occurs when the initial value of the L / norm is smaller than a certain critical value. In the recent [DP4] it is prove that the critical value 8π of the initial mass prouces an optimal threshol between blow up an global existence when =. The situation in > is not fully unerstoo, an there is a possible range of choice of initial ata where both blow-up an existence can occur. We refer to the introuction of [CPZ4] for a etaile escription of the subject. An interesting question, in this framework, is whether is possible to give sense to the evolution of the moel after the blow up. This problem is stuie in [Vel, Vel4]. We mention here (an refer the intereste reaer to) the survey papers [Hor3, Hor4], where more general chemotaxis moels are presente, together with an extensive list of results an references. Although of great mathematical interest, moels allowing for the infinite growth of solutions have often been criticize because their biological interpretation is not

3 CHEMOTAXIS WITH PREVENTION OF OVERCROWDING 3 fully unerstoo. As a consequence of that, several generalizations of the Keller Segel moel, where the formation of singularities is prevente a priorily, have been formulate recently. The basic assumption in these moels is the existence of a maximal value for the cell ensity, which is reasonable in certain physical situations (see the introuction of [HP]) where cells stop aggregating after a certain size of the aggregate has been reache. In [HP], a chemotaxis moel featuring a nonlinear flux term is presente, where the chemotactic response is shut off when a certain cell ensity has been reache. This moel, being of type (.5), but with a parabolic equation for S, prevents the overcrowing of cells, an the authors prove global existence of solutions uner Neumann bounary conitions. In a secon paper [PH3], the authors erive this moel from a master equation escribing a biase ranom walk on the real line, assuming that the probability of a cell moving to the right or the left epens on the ifference in chemical concentration as well as on the cell ensity. The resulting evolution law for the ensity is the first equation of (.) with mobility M(ρ) = ρ( ρ) an logarithmic entropy for ρ in the energy (although there is no clear separation between energy an mobility in [PH3], see also the consierations below on the relation between ranom mobility an chemotactic sensitivity). Note that the main ifference to a stanar erivation of the moel without prevention of overcrowing is the aitional factor ( ρ) in the mobility an the aitional entropy term epening on ( ρ) in the energy (see also [Wrz4a, Wrz4b], where the same entropy is use to stuy the asymptotic behaviour in the fully parabolic case in a boune omain). Intuitively, this change of mobility seems obvious, since the possibility of cells to move freely is limite by the cells aroun. Therefore, in the case of overcrowing (which happens in our scaling for ρ ), their motion woul be stoppe. The aitional entropy term epening on ( ρ) is less obvious to interpret, it somehow forces cells to iffuse even in the case of overcrowing. On the other han if the volume filling mechanism is thought of as a finite size effect really blocking the cells, then also no iffusion woul be allowe an the effect woul have consequences only upon mobility but not on the energy. This argument is one reason why we shall also iscuss a ifferent choice in the energy in this paper. More precisely, we shall use a quaratic energy of the form E(ρ, S) = ε ρ x ρs x + ( S + S ) x, (.6) R R R yieling the egenerate parabolic-elliptic problem ρ = (ρ( ρ) (ερ S)) t (.7) S + S = ρ. Several reformulations of the Keller Segel moel have been stuie recently, with a nonlinear iffusion term in the evolution of the cell ensity replacing the linear one. For an overview of the biological motivations an the existence theory for such moels we refer to [Hor3, Chapter6]. An interesting issue is whether is possible to avoi blow up of solutions by introucing nonlinear iffusion effects without the help of the volume filling term in the mobility. A stuy towars this irection has been recently carrie out in [Kow5]. In [CC5] a simple threshol conition for the nonlinear iffusion term of the ensity is establishe in orer to achieve global existence in time (in the special case where the balance law of the cell ensity is

4 4 M. BURGER, M. DI FRANCESCO, AND Y. DOLAK couple with Poisson equation). We stress that the main feature in our nonlinear moel (.7) is that the ranom mobility (i. e. the nonlinear iffusion coefficient) is vanishing at the threshol value ρ =. Such a property is not usually covere in the literature concerning chemotaxis moels with nonlinear iffusion. In aition to the remarks mae above concerning the interplay between energy an mobility, another motivation for our choice of the system (.7) is a simplification of an alternative formulation of the Keller-Segel moel recently erive in [BO4]. Due to the physical observation that many chemicals appear both to stimulate irecte motion up the chemotactic graient an to alter the ranom mobility coefficient, the authors of [BO4] introuce a moel base upon a multiphase interpretation of the cell ensity (in an incompressible material compose of the phases cells an water), where a irect relationship between the chemotaxis an ranom motion coefficients is establishe. In particular, they both may vanish at the threshol value of the ensity. A stuy of the well poseness theory for moels of that type has been starte in [LW4], in the fully parabolic case on boune omains, where the existence of nontrivial stationary solutions has been proven. Using an interpretation as a multiphase system, it is natural to make the connection to Cahn-Hilliar equations (cf. [CH58, CH7]), which have the same structure as the first equation in (.), but with the potential µ etermine as µ = ρ + W (ρ) for some ouble-well potential function W having minimizers at zero an one. The prevention of overcrowing by a mobility M(ρ) = ρ( ρ) in the Cahn-Hilliar equation is well motivate from physical arguments an has been stuie in etail (cf. [CENC96, CT94, EG96]). Another argument in favour of (.7) relies on the asymptotic behaviour, in particular on the existence both of ecaying an of non-ecaying solutions of (.7). We recall that the long-time asymptotics of (.5) in boune intervals (uner Neumann bounary conitions) have been recently stuie in [DS5] an, with a parabolic equation for S, in [PH5]. The observe behaviour is a coarsening process reminiscent of phase change moels, where plateau-like peaks of the cell ensity form after a short transition perio an then merge exponentially slowly. Numerical stuies inicate that in most situations the only stable stationary states are single plateaus locate at bounary of the omain. It is therefore not surprising that the behaviour of solutions on the whole space is ifferent. Roughly speaking, the cells are not stoppe by any bounaries an therefore the linear moel woul allow them to sprea out rather than aggregate. We shall make this statement more rigorous by several results on the ecay of the cell ensity ρ for large time. For the present moment we emphasize that, to our knowlege, the choice of the system (.7) is the only known version that achieves stationary profiles on the whole space. Nonlinear egenerate iffusion moels have been use to escribe various biological phenomena such as the ispersal of biological populations (cf. e.g. [GM77]) or aggregation of animal population (calle swarming, cf. e.g. [NM83, MCO5, MEK99, TB4]), the latter exhibiting many further analogies to moels for chemotaxis. In these applications, as well as in the moel consiere here, the main avantages of nonlinear iffusions are a finite spee of propagation (which seems more reasonable than infinite spee in particular in biological applications) an, as mentione above, the existence of nontrivial stationary an non-ecaying timeepenent solutions.

5 CHEMOTAXIS WITH PREVENTION OF OVERCROWDING 5 In orer to make the comparison between the linear an nonlinear iffusion case more concrete, we give an overview of the main results of this paper in the following, which also provies a guieline through the paper for reaers rather intereste on the results than on the etaile proofs. Concerning existence, we have Global existence an uniqueness of weak solutions in the case of linear iffusion (Section.). Global existence of weak an entropy solutions, uniqueness of entropy solutions of boune variations in the case of nonlinear iffusion (Sections 3. an 3.) an finite spee of propagation of the support in (Section 3.3). The first step towars the large time behaviour of solutions is an investigation of possible stationary solutions. Here we fin a first ifference between the two cases, namely Non-existence of finite mass stationary solutions ifferent from zero for linear iffusion (Subsection..). Existence of finite mass stationary solutions ifferent from zero for nonlinear iffusion in the case ε < (Subsection 3.4.). A partly numerical construction of stationary solutions even inicates their existence for arbitrary mass if ε < (Section 3.5). The etaile large-time behaviour is escribe by: Decay to zero of solutions in the linear iffusion case in the following two cases: with arbitrary initial mass for ε > 4 in (Section.), with small initial mass for arbitrary ε > (Section.). In both cases, solutions converge in L towars the self similar Gaussian solution of the heat equation with variance ε (Section.3). Existence of non-ecaying solutions for ε < in the nonlinear case (Subsection 3.4.), these solutions converge (along subsequences) to stationary solutions (Subsection 3.4.). For ε > all compactly supporte solutions ecay an their support must become unboune as t (Subsection 3.4.3). The result summarize in the last item above, namely the possibility of achieving two rastically ifferent asymptotic behaviours (i. e. convergence to nontrivial stationary solutions or ecay to zero) by simply changing the iffusivity constant ε in (.7) is the main result of our paper. We stress here that the behaviour of solutions for ε > (i. e. their ecay to zero) is inepenent of any initial parameter such as the total mass or the secon moment. This fact constitutes an essential ifference between (.7) an classical moels with linear iffusion on the whole R. The main technique use throughout the paper is the use of suitable energy functionals or Lyapunov functionals, which are alreay known to be extremely helpful in orer to achieve global existence for general nonlinear Keller Segel moels (see [Hor3]).. Linear iffusion In this section, we first cover the existence an uniqueness theory for weak solutions of moel (.5). We then turn our attention to the asymptotic behaviour for

6 6 M. BURGER, M. DI FRANCESCO, AND Y. DOLAK large time. The local existence an uniqueness is achieve by means of a stanar fixe point technique. The continuation for any time of the solutions is a ue to the existence of the invariant omain ρ, which is consistent with the volume filling assumption iscusse in the introuction. Similar results are presente in case of boune omains in [HP] (for the fully parabolic moel) an in [DS5]... Existence theory an preliminaries. We stuy the parabolic elliptic system { ρ t = ε ρ (ρ( ρ) S) (.) S + S = ρ, where x R,, t, ε >, ρ an S are scalar functions. We recall that the above system can be ecouple in orer to get a nonlocal parabolic equation for ρ by means of the convolution representation formula S(x, t) = B(x y)ρ(y, t)y, R where B is the Bessel potential + x t e 4t B(x) = t. (.) (4π) / t /... Existence of solutions. Let us consier the Cauchy problem { ρ t = ε ρ (ρ( ρ) (B ρ)) ρ(x, ) = ρ (x). (.3) We shall establish a local in time existence result for (.3). First, let us recall some elementary properties of the Bessel potential (.) an of the heat kernel G(x, t) = x e 4εt (4πεt) /, x R, t >. (.4) The proof of the following lemma follows by straightforwar computations. Lemma.. The following estimates are satisfie, B L (R ) = (.5) B L (R ) < + (.6) G(, t) L p (R ) Ct (p ) p, p (.7) G(, t) L p (R ) Ct (p ) p, p (.8) Theorem. (Local existence). Let ρ L (R ) L (R ) (resp. ρ L (R )). Then, there exists a unique solution ρ(x, t) to (.3) such that ρ L ( [, T ], L (R ) L (R ) ) (resp. ρ L ( [, T ], L (R ) ) ) for a small enough positive time T. Proof. Let is first assume ρ L (R ) L (R ). Given M, T positive constants, for ρ L ( [, T ], L (R ) L (R ) ) we efine the norm [ ] ρ T := ρ(t) G(t) ρ L (R ) + ρ(t) G(t) ρ L (R ), sup t T

7 CHEMOTAXIS WITH PREVENTION OF OVERCROWDING 7 the Banach space X M T := an the map ρ T ρ (T ρ) (x, t) = (G ρ ) (x, t) + { ρ L ( [, T ], L (R ) L (R ) ) } ρ T M, t R G(x y, s t) (ρ( ρ) (B ρ)) (y, s)sy. Thanks to the Duhamel principle for the linear heat equation (an thanks to integration by parts), a fixe point for T is clearly a weak solution to the Cauchy problem (.3) with initial atum ρ on the strip [, T ] R. To establish the existence an uniqueness of the fixe point we first prove that T ρ XT M, then we prove that T : XT M X T M is a strict contraction, an we euce the esire assertion by the Banach fixe point theorem. Let ρ XT M. Young inequality for convolutions, together with (.6), (.7) an (.8) yiel, for any < t < T, (T ρ)(t) G(t) ρ L (R ) C C t t t G(t s) L (ρ( ρ) (B ρ))(s) L s (t s) / ρ( ρ)(s) L B L ρ(s) L s (t s) / ( ρ(s) L + ρ(s) L ) ρ(s) L s C(M + ρ L )(M + ρ L + )(M + ρ L )T / Analogously, (T ρ)(t) G(t) ρ L (R ) C t t G(t s) L (ρ( ρ) (B ρ))(s) L s (t s) / ρ( ρ)(s) L B L ρ(s) L s C(M + ρ L )(M + ρ L + )(M + ρ L )T / The two estimates above ensure T ρ XT M provie T is small enough. We now consier two elements ρ, ρ XT M. For any < t < T, in a similar fashion as above we have (T ρ)(t) (T ρ)(t) L (R ) + + t t t t G(t s) L (ρ( ρ) (B ρ))(s) (ρ( ρ) (B ρ))(s) L s G(t s) L (ρ( ρ) (B ρ))(s) (ρ( ρ) (B ρ))(s) L s G(t s) L (ρ( ρ) (B ρ))(s) (ρ( ρ) (B ρ))(s) L s G(t s) L (ρ( ρ) (B ρ))(s) (ρ( ρ) (B ρ))(s) L s C( ρ L, M)T / sup <t<t ρ(t) ρ(t) L, (.9)

8 8 M. BURGER, M. DI FRANCESCO, AND Y. DOLAK an, analogously (T ρ)(t) (T ρ)(t) L (R ) C( ρ L, M)T / Hence, by choosing T small, we have T ρ T ρ T α ρ ρ T, sup <t<t ρ(t) ρ(t) L. for some < α <, which conclues the proof in case ρ L (R ) L (R ). The proof in case ρ L (R ) can be obtaine by moifying the norm T an the Banach space XT M (by taking into account the L norm only) an by repeating the same steps as in the previous case. Remark.3 (Continuation principle). From the implicit representation formula t ρ(x, t) = (G ρ ) (x, t) + G(x y, t s) (ρ( ρ) (B ρ)) (y, s)sy R (.) of the local solution ρ, we also euce that, if T M is the maximal time of existence of the solution, then there exist t j T M as j such that lim [ ρ(t j) L + ρ(t j ) L ] = +. j Remark.4 (Higher regularity). The local in time solution ρ(x, t) provie by theorem. is enowe with (at least) the same regularity with respect to x of the initial atum. To see this, one can moify the space an the norm in the proof of the previous theorem by taking into account some L p norm of ρ. A bit of regularity for ρ without the help of any further requirements on the initial atum can be obtaine at least in. Proposition.5 (Regularizing effect). Let the initial atum ρ L (R) L (R). Then, at any positive time t, the solution ρ(t) of (.) is continuous with respect to x. Proof. Since G ρ is a C function, we only nee to prove that ρ G ρ is continuous. By formula (.), we have for small h > (ρ G ρ )(x + h, t) (ρ G ρ )(x, t) = t + [G x (x + h y, t s) G x (x y, t s)] (ρ( ρ) (B ρ)) (y, s)sy. We recall that the term (ρ( ρ) (B ρ)) is (locally) boune. Moreover, thanks to (.8) we can fin nonnegative functions H(x, y, t, s) an K L such that ( ) x y G x (x + h y, t s) G x (x y, t s) H(x, y, t, s) (t s) K. t s Therefore, the limit as h of the left han sie above is zero in view of Lebesgue s ominate convergence theorem. Theorem.6 (Continuity with respect to the initial ata). Let ρ an ρ two local in time solutions to (.) with initial ata ρ, ρ L L respectively. Then, for a small T we have ρ(t) ρ(t) L C(T ) ρ ρ L, (.) for all t [, T ].

9 CHEMOTAXIS WITH PREVENTION OF OVERCROWDING 9 Proof. The proof can be performe by means of a similar argument as in (.9). Since, in this case, we have two ifferent initial ata, we easily obtain ρ(t) ρ(t) L CT / sup ρ(t) ρ(t) L + ρ ρ L, t T where C epens on the two solutions. For small T we have the esire estimate (.).... Global existence. In the following, we will be intereste in initial ata ρ in (.3) satisfying the assumption ρ (x), x R. (.) Our aim is to prove that conition (.) is invariant uner the flow inuce by the moel (.). Similar properties are proven in [HP] for the fully parabolic moel an in [DS5] in a boune omain. Let us start with the following theorem. Theorem.7 (Conservation of the total mass). Let ρ L (R ) satisfying (.). Then, ρ(x, t)x = ρ (x)x. (.3) R R Proof. Let ζ n = ζ n (x) be a sequence of cut off functions such that ζ n (x) = as x n, ζ n (x) = as x n+, ζ n (x) as n x n+ an ζ n C (R ). Multiplying the equation in (.3) by ζ n, after integration over R [, T ] we obtain T ρ(x, t)ζ n x ρ (x)ζ n x = ρ t ζ n xt R R R T = ζ n [ε ρ (ρ( ρ) S)]xt R T T = ε ζ n ρ + ρ( ρ) S ζ n xt as n + R R The last step is justifie by ominate convergence theorem. The whole calculations are justifie in case of smooth solutions. Hence, we shall assume that the initial atum belongs in some suitable Sobolev space (see remark.4). The result for a general initial atum ρ then follows by approximating ρ with a sequence of smooth initial ata an by using the continuity result in theorem.6 an Fatou s lemma. Theorem.8 (Global existence). Assume the initial atum ρ L (R ) satisfies (.). Then, there exists a unique global solution to the Cauchy problem (.3), satisfying ρ(x, t) for any (x, t) R [, ). (.4) In particular, < ρ(x, t) < if < ρ (x) <. (.5) Proof. Writing (.3) as ρ t + ρ S( ρ) + ρ( ρ)(s ρ) = ε ρ, it can be seen immeiately that ρ an ρ are lower an upper solutions, respectively. By the mean value theorem of multiimensional calculus, the function w(x, t) = ρ ρ satisfies the inequality w t + A(x, t) w + B(x, t)w ε w,

10 M. BURGER, M. DI FRANCESCO, AND Y. DOLAK with boune coefficients A(x, t) an B(x, t). For w(x, t) = ρ ρ, the same equations with a reverse inequality sign hols, an the bouneness of ρ follows from the Phragmèn-Linelöf principle for parabolic equations [PW84]. Together with theorem.7 an with remark.3, this implies the assertion... Decay of solutions as t +. In this subsection we etermine sufficient conitions on the iffusivity constant ε in (.) an on the initial atum ρ such that the L (R ) norm of the corresponing solution ρ(x, t) tens to zero as t +. When such a phenomenon occurs, the iffusion term ε ρ in (.) becomes ominant. We prove here that this is the case when either the mass is small enough for arbitrary ε > or for any mass in case ε > /4 in. It is alreay known in case of moels without prevention of overcrowing that when the mass of the initial atum is much smaller than some constant epening on ε, then the iffusion term becomes ominant an prouces a long time ecay of the solution (see [CPZ4, DP4]). For the sake of completeness we shall reprouce the same result for our moel in the following proposition, the proof of which being partly the same as in [CPZ4]. Proposition.9. Let ε >. Let ρ L (R ) satisfying (.). Then, there exists a constant C(, ε) epening only on the imension an on the iffusivity ε, such that, if the total mass satisfies R ρ x < C(, ε), (.6) then, the solution ρ(x, t) to (.3) satisfies the ecay estimates ρ(t) L p (R ) C(t + ) (p ) p, p. (.7) Proof. Let us start with the L p estimates for finite p. By multiplying the equation in (.3) by pρ p an after integration by parts we obtain ρ p (x, t)x = εp ρ p ρx + p(p ) (ρ p ρ p ) S ρx t R R R 4(p ) ε ρ p/ x + (p ) ρ p Sx p R R 4(p ) ε ρ p/ x + (p ) ρ p+ x, p R R where we have use the a priori estimate ρ an S. By means of the Gagliaro Nirenberg inequality ρ p+ x C(p, ) ρ L α (R ) ρ p/ x, R R where α = for =,, α = / for >, we easily get ( ) 4ε ρ p (x, t)x (p ) C(p, ) ρ x ρ p/ x t R p R R Hence, for R ρ x < t 4ε pc(p,), we can write, for some C >, ρ p (x, t)x + C ρ p/ x. R R

11 CHEMOTAXIS WITH PREVENTION OF OVERCROWDING Thanks to the following interpolation inequality (see e.g. [EZ9]) ((p )+)p (p ) ρ C(p, ) ρ p/ p L p (R ) L (R ) ρ (p ) L (R ) we have ( ) (p )+ ρ p (x, t)x + Cm ρ p (p ) x t R R which implies the esire polynomial time ecay in L p ρ(t) L p (R ) C(p, )(t + ) (p )/p (.8) where the (t + ) instea of t is justifie by the global in time control of all the L p norms proven in the previous subsection. In orer to obtain the corresponing L estimate, we employ the implicit representation of the solution ρ provie by Duhamel principle ρ(x, t) = G(t) ρ(t) + = G(t) ρ(t) + t t By taking the L norm in (.9), we obtain ρ(t) L (R ) G(t) L (R ) ρ(t) L (R ) + t t G(t σ) (ρ( ρ) B ρ)(σ)σ G(t s) (ρ( ρ) B ρ)(t + s)s. (.9) G(t s) L r (R ) ρ( ρ) B ρ(t + s) L r (R ) s t Ct + C (t s) (r ) r ρ(t + s) L r (R ) s t Ct + (t s) (r ) r (t + s) (r ) r s = Ct + Ct for r <, r = r r. Of course, since ρ(t) L (R ) is uniformly boune an since, we have the estimate ρ(t) L (R ) C(t + ). In the next proposition we prove that solutions to (.3) enjoy a time ecay rate as in (.7) no matter how large the mass is, provie ε > /4 an =. This result constitutes an essential ifference of the present moel with respect to the classical moels where overcrowing phenomena (i. e. blow up) may occur in finite time. Proposition.. Let ε > /4 an =. Let ρ L (R) satisfying (.). Then, the solution ρ(x, t) to (.3) satisfies the ecay estimates ρ(t) L p (R) C(t + ) (p ) p, p. (.) Proof. We start by performing the following L estimate. ρ (x, t)x = ε ρ ρx + ρ( ρ) S ρx t R R R ε ρ x + ρ S x R R

12 M. BURGER, M. DI FRANCESCO, AND Y. DOLAK As a consequence of the Young inequality for convolutions we have ρ S x ρ x R R an therefore ( ρ (x, t)x ε ) ρ x. t R 4 R Then, by means of the Gagliaro Nirenberg inequality, as in the previous proposition, we get the ecay in L ρ(t) L(R) C(t + ) /4. By taking the L 4 norm in the representation (.9), in a similar fashion as in the previous proposition, we obtain Finally, ρ(t) L 4 (R) C(t + ) 3/8 + C(t + ) 3/8 + t ρ(t) L (R) C(t + ) / + C(t + ) / + t t G(t s) L 4 ρ B ρ(t + s) L s (t s) 7/8 (t + s) / s C(t + ) 3/8 + Ct 3/8. t G(t s) L ρ B ρ(t + s) L s (t s) 3/4 (t + s) 3/4 s C(t + ) /. The remaining L p estimates are easily obtaine by interpolation.... Some remarks an the nonexistence of stationary solutions. Clearly, an open question is whether solutions to (.3) ecay for any ε an for arbitrarily large masses. In boune omains, for = an Neumann bounary conitions, system (.) has been shown to ecay to the constant solution if ε > 4, but if ε is small enough, stationary, perioic solutions in L ((, L)) exist (see [DS5] an [PH5]). However, one can easily prove that there exist no nonzero stationary solutions to (.) in L (R ) in the case of unboune omains: We efine the energy functional E(ρ, S) of system (.) by E = ( S + S )x with E ρ = S + ε log ρ ( ρ) Rewriting the first equation of (.) as ρ t = ρsx + ε [ρ log ρ + ( ρ) log( ρ)]x, (.) an ( ρ( ρ) E ρ E S =. (.) ), (.3) an ifferentiating the energy with respect to time, we obtain E t = E S S t + E ( ρ ρ t = ρ( ρ) E ) E ρ ρ x = ρ( ρ) E x. ρ

13 CHEMOTAXIS WITH PREVENTION OF OVERCROWDING 3 Hence, the energy is ecreasing in time, an the stationary state E t = is only reache if ρ =, ρ = or E ρ = const., the latter implying that the stationary solution (ρ, S) shoul satisfy, for some positive constant C, ρ ρ = e S+C ε, in some open set R, with ρ = in some point of. However, this is incompatible with S being boune, because of the continuity of ρ state in proposition Self similar long time behaviour. The aim of this subsection is to prove that, uner suitable assumptions on the initial atum ρ, the solution to (.3) behaves asymptotically like the funamental solution of the heat equation. To perform this task we employ the entropy issipation metho (see [AMTU, CT]). The long time ecay properties of the solution ρ(x, t) are a crucial ingreient in the arguments below. We shall prove our result by assuming a priori that the solution ρ(x, t) satisfies the ecay estimate ρ(t) L (R ) C(t + ) (.4) which we prove to be fulfille uner the assumptions in propositions.9 an Preliminaries. Our first step is the following stanar (mass preserving) time epenent rescaling ρ(x, t) = R(t) v(y, s) y = R(t) x s = (.5) log R(t) R(t) = t +. Then, it is easily seen that v(y, s) satisfies the Cauchy problem { v s = (ε v + yv) e s (v( e s v)b s v) v(y, ) = ρ (y) (.6) where B s (y) = e s B(e s y). For future reference we write equation (.6) as follows ( )) v s = ε v (log v + y e s (v( e s v)b s v). (.7) ε We remark that the funamental solution G(x, t) of the heat equation in rescale variables epens only on y; more precisely, it is given by U m (y) = Ce y ε, where C epens on the mass m of ρ. Moreover, we recall that U m satisfies the elliptic equation (ε U m + yu m ) =. We shall make use of the classical entropy functional E(v) = v(y) log v(y)y + y v(y)y, (.8) R ε R an of the relative entropy RE(v U m ) = E(v) E(U m ).

14 4 M. BURGER, M. DI FRANCESCO, AND Y. DOLAK We observe that, thanks to the alternative form (.7), equation (.6) can be written as ( v s = ε v δe(v) ) e s (v( e s v)b s v), (.9) δv where δe δv is the first variation of the functional E. Once the mass m is fixe, the relative entropy functional attains zero as minimum value at the groun state U m (see [CT, MV]). The following inequality (see [AMTU] for the proof) establishes a connection between the convergence in relative entropy an the convergence in L. Theorem. (Csiszár Kullback inequality). Let v L (R ) having mass m an such that E(v) < +. Then, there exists a fixe constant C (epening on m) such that v U m CRE(v U m ). (.3) For future reference, we recall the following logarithmic Sobolev inequality ue to Gross [Gro75] an subsequently generalize in [AMTU]. Theorem. (Logarithmic Sobolev inequality). Let v L (R ) having mass m an such that E(v) < +. Then, the following inequality is satisfie where I(v U m ) = R v RE(v U m ) ε I(v U m), (.3) ( log v U m is calle (relative) Fisher information. ) y = R v ( ) log v + y y ε Finally, we remark that assumption (.4) for ρ in the new variables v(y, s) reas v(s) L (R ) C (.3) for some fixe C > epening only on the initial atum..3.. Tren to self similarity. In this subsection we employ the tools introuce above to prove the asymptotic self similar behaviour of solutions satisfying (.4). Theorem.3. Let ρ L (R ) satisfying (.) an such that E(ρ ) <, where E(ρ ) is efine in (.8). Suppose that the corresponing solution ρ(x, t) to (.3) satisfies the time ecay conition (.4). Then, { o(t +δ ) for arbitrary < δ if = ρ(t) G(t) L (R ) = O(t (.33) ) if >, where G is the funamental solution of the heat equation (.4) with same mass as ρ. Proof. In what follows, we shall enote a generic positive constant inepenent on s by C. Let us multiply equation (.6) by log v(y) + y ε an integrate over R.

15 CHEMOTAXIS WITH PREVENTION OF OVERCROWDING 5 Then, integration by parts an conservation of the total mass yiel ( s E(v(s) U m) = ε v log v ) ) y + Ce s v B s v (log v + y y R U m R ε ) ( / ( ) ) εi(v U m ) + Ce (R s v B s v y log v + y / y ε =: εi(v U m ) + J. (.34) We now estimate the term J by using (.3) an Young s inequality. ( ) / J Ce s R v v y I(v U m ) / = Ce (I(v U s m ) + v ) / v y y I(v U m ) / R ε R Ce s ( I(v U m ) + I(v U m ) /). R v Therefore, for fixe C, C >, inequality (.34) implies, s E(v(s) U m) (ε C e s )I(v U m ) + C e s I(v U m ) / an, for an arbitrarily small δ >, s E(v(s) U m) (ε Ce s Ce δs )I(v U m ) + Ce s+δs. For s s (ε) we have ε Ce s Ce δs > an we can use inequality (.3) to obtain s E(v(s) U m) (ε Ce s Ce δs ) E(v(s) U m ) + Ce s+δs. ε Hence, thanks to the variation of constants formula we obtain the following ecay rates as s + (here < δ ) E(v(s) U m ) = { O(e ( δ)s ) if = O(e s ) if >. (.35) Finally, by means of the Csiszár Kullback inequality (.3) an by using the original variables ρ(x, t) accoring to (.5), we obtain the esire estimate (.33). Remark.4. At the en of their work, the author have been informe that J. Dolbeault an A. Blanchel have proven a similar result concerning self similar asymptotic behaviour for Keller Segel moel without the volume filling effect where the evolution of S is escribe by Poisson s equation. 3. Nonlinear iffusion In this section we focus our attention on the nonlinear moel ρ = (ρ( ρ) (ερ S)) t S + S = ρ, subject to the initial conition (3.) ρ(x, ) = ρ (x), ρ L (R ), ρ (x), x R. (3.)

16 6 M. BURGER, M. DI FRANCESCO, AND Y. DOLAK 3.. Existence of Weak Solutions. We start by proviing a suitable efinition of weak solutions. In orer to simplify the notation, we fix ε = throughout this section, the value of ε not being relevant in the existence theory. We enote A(ρ) = ρ ξ( ξ)ξ = ρ ρ3 3, A(ρ) = ρ A(ξ)ξ = ρ3 6 ρ4. Definition 3.. A function ρ L ([, + ) R ) is calle a weak solution of the Cauchy problem (3.) (3.) on R [, T ] (T eventually + ) if the following conitions are satisfie, (i) A(ρ) L ([, T ]; L (R )) (ii) A(ρ) L ([, T ] R ) (iii) ρ(x, t) almost everywhere in [, T ] R. (iv) For all ϕ Cc ([, T ) R )., the following relation hols T T ρϕ t xt A(ρ) ϕxt R R T + ρ( ρ) S ϕxt + ρ(x, )ϕ(x, )x =, R R where S = S[ρ] is the unique H solution to S + S = ρ. In orer to prove local in time existence of weak solutions to the Cauchy problem (3.) (3.), we use the following strategy. We first regularize the parabolic equation in (3.) by aing a small linear iffusion term an we solve the Cauchy Dirichlet problem on a boune omain via a priori estimates an Schauer s fixe point theorem. At this level, we prove that the conition ρ is preserve. We then exten the result to the Cauchy problem (3.) (3.) by a compactness argument. Such a proceure is inspire by a similar argument in [BCM3]. The main technical problem in our case relies on the egeneracy of the nonlinear iffusion coefficient at the threshol value ρ =, which reners existence a non trivial issue. We overcome this ifficulty by incluing the invariant omain property ρ in our efinition of solutions. This property prevents the overrunning of the mobility coefficient ρ( ρ) into the backwar iffusion range. Remark 3.. Unlike the notion of weak solution given in efinition 3. for problem (3.) (3.), we shall usually refer to an L istributional solution of any of the various approximating problems treate below as a weak solution Non egenerate approximation in a boune omain. Let R be a boune omain with smooth bounary. For fixe T > we enote as usual T = [, T ]. For small µ, ν >, we stuy the approximating IBV problem ρ t = (a µ,ν(ρ) ρ b ν (ρ) S) as (x, t) T S + S = ρ, as (x, t) T, (3.3) ρ(x, t) = S(x, t) = as x, ρ(x, ) = ρ (x), where a µ,ν (ρ) = µ + b ν (ρ) an b ν (ρ) is a uniformly boune (with respect to ν), smooth approximation of the function ρ R [ρ( ρ)] +

17 CHEMOTAXIS WITH PREVENTION OF OVERCROWDING 7 as ν +, such that b ν (ρ) for all ρ an b ν (ρ) > if an only if ρ (, ). In orer to prove existence of solutions for small T, we first efine the operator L ([, T ]; H ()) S U(S) as follows ρ t = (a µ,ν(ρ) ρ b ν (ρ) S) as (x, t) T U(S) = ρ where ρ solves ρ(x, t) = S(x, t) = as x, ρ(x, ) = ρ (x). (3.4) For a given S L ([, T ]; H ()), the existence an the uniqueness of a weak (smooth) solution to (3.4) is guarantee by the smoothness of the iffusion coefficient a µ (ρ) an by a µ (ρ) µ > (see [LSU67], Chapter V, Subsection 6, Theorem 6.). Moreover, it is easily checke that the operator U : L ([, T ]; H ()) L ( T ) is well efine. Then, we enote by D : L () H () the solution operator of the elliptic equation S + S = ρ, (3.5) more precisely we set D(ρ) = S. Finally, we set T = U D. In what follows we recover some properties of the maps U an D neee to apply Schauer s fixe point theorem. Proposition 3.3. The map U : L ([, T ]; H ()) L ( T ) is continuous an compact. Proof. We enote by : H () H () the inverse of the Laplacian on ρ with Dirichlet bounary conitions. Multiplication of equation (3.4) by t an integration by parts over yiel A µ,ν (ρ(t))x + ρ t ρ t x = b ν (ρ) S t x sup b ν (ρ) S ρ ρ t x, where A µ,ν (z) = z ξ a µ,ν(ζ)ζξ. Hence, by Young s inequality we obtain A µ,ν (ρ(t))x + ρ t t x ( ) sup b ν (ρ) S x, (3.6) ρ an, after integration on [, T ], T ρ t x C T S x + A µ,ν (ρ )x. (3.7) Moreover, multiplication of (3.4) by A µ,ν (ρ) := ρ a µ,ν(ζ)ζ an integration by parts imply t A µ,ν (ρ(t))x + A µ,ν (ρ) x = b ν (ρ) S A µ,ν (ρ)x. (3.8)

18 8 M. BURGER, M. DI FRANCESCO, AND Y. DOLAK By means of Young s inequality one can manipulate (3.8) in a similar fashion as before an obtain, after ouble integration on [, T ], T T A µ,ν (ρ(t))x + A µ,ν (ρ) xt T T C A µ,ν (ρ )x + C S xt. (3.9) The efinition of a µ,ν implies ρ C(µ)A µ,ν (ρ), A µ,ν(ρ) µ. Therefore, inequalities (3.9) an (3.7) imply the following statement ρ L ([,T ];H ()) + t ρ L ([,T ];H ()) C(µ, T ) [ S L ([,T ];H ()) + C(ρ ) ], (3.) where we have use the fact that : H () H () is an isomorphism. Let us now take a sequence {S n } n uniformly boune in L ([, T ]; H ()) an consier the corresponing ρ n = U(S n ). Inequality (3.) an an embeing result by Lions an Aubin (see for instance [Sho97], Chapter III, Section, Proposition.3) imply that {ρ n } is compact in L ( T ). To prove continuity, take a sequence S n L ([, T ]; H ()) such that S n S. Then, by compactness of U, U(S n ) = ρ n has a convergent subsequence in L ( T ). Therefore, ρ n has a subsequence ρ nk converging almost everywhere to some ρ L ( T ). By using the weak formulation of problem (3.4) with S = S nk an ρ = ρ nk, we can easily euce that ρ is the unique weak solution to (3.4) where S = S. Therefore, the whole sequence ρ n converges to U( S) an the proof is complete. For the sake of completeness, we provie the etails about the continuity of D, which is of course straightforwar. Proposition 3.4. The (linear) operator D : L ( T ) H ( T ) is continuous. Proof. It follows from the estimate S(t) H () ρ(t) L (), (3.) which can be easily proven by multiplying equation (3.5) by S an by integrating over. Theorem 3.5. There exists at least one local in time solution of problem (3.3). Proof. Relation (3.9) clearly implies that ρ L ( T ) C(µ) T [ ] C(ρ ) + S L ([,T ];H ()) an this, together with (3.), implies ρ L ( T ) C(µ) T [ C(ρ ) + ρ L ( T )]. (3.) Hence, for M > we can consier the Banach space { } X M = ρ L ( T ) ρ xt M. T Estimate (3.), then, clearly implies that T = U D is well efine as a map from X M into itself provie that T is small enough an M large enough. Moreover, thanks to propositions 3.3 an 3.4, T : L ( T ) L ( T ) is continuous an

19 CHEMOTAXIS WITH PREVENTION OF OVERCROWDING 9 compact. Therefore, T has a fixe point thanks to Schauer s fixe point Theorem (see e.g. [Tay97], Chapter 4, Corollary B.3), which means that (3.3) has a local in time solution ρ L ( T ). Remark 3.6. A continuation principle in the spirit of remark.3 hols in this case too, where the L an the L norms are replace by the L norm. Moreover, by integrating once with respect to time in the ientity (3.8), one easily gets an estimate of the local solution ρ in L ([, T ], L ()). Our next aim is to prove that conition (3.) is preserve by the local solution ρ(t) of problem (3.3) at any t [, T ]. Lemma 3.7. Let ρ(x, t) be the local solution to (3.3) provie by Theorem 3.5, with initial atum ρ satisfying (3.). Then, for any t T, we have ρ(x, t), a. e. in x. Proof. We claim that ρ(x, t) can be obtaine as a uniform limit of the sequence ρ n (x, t) recursively efine by ρ t = (a µ,ν(ρ n ) ρ b ν (ρ) S) as (x, t) T ρ n solution of S + S = ρ, as (x, t) T, ρ(x, t) = S(x, t) = as x, ρ(x, ) = ρ (x), where the local existence for the above equation can be proven in the same way as before (the iffusion term is linear!). To see this, one can apply the argument in the proof of proposition 3.3 to the semi linear equation satisfie by ρ n an get the estimate (3.) uniformly in n. Then, in the same way we get compactness in L ( T ) of the family ρ n, an this is enough to get a weak solution to the nonlinear equation (3.3) in the limit as n. Now, by means of the same comparison argument as in Theorem.8, we get the esire estimates for any n (the parabolic part of the operator is linear an nonegenerate), an the same relation hols as n. Remark 3.8. As a consequence of the previous lemma an thanks to the continuation principle 3.6, the local in time solution to (3.3) is actually globally efine. Remark 3.9. Higher regularity of the local solution when the initial atum ρ is smooth enough can be proven by moifying the fixe point argument, involving higher Sobolev norms an exploiting the regularizing properties of the solution operator S of the elliptic equation S+S = ρ, the nonegeneracy of the parabolic equation in (3.3) an the result in Lemma 3.7 (we omit the etails). Our next step is the limit as ν in (3.3). We have the following theorem. Theorem 3. (Existence for the nonegenerate approximation). For any small µ >, there exists at least one global in time weak solution to the problem ρ t = (a µ(ρ) ρ [ρ( ρ)] + S) as (x, t) T S + S = ρ, as (x, t) T, (3.3) ρ(x, t) = as x, ρ(x, ) = ρ (x),

20 M. BURGER, M. DI FRANCESCO, AND Y. DOLAK where a µ = µ + [ρ( ρ)] +, with ρ satisfying (3.). Moreover, the local in time solution ρ satisfies ρ(x, t) (3.4) almost everywhere. Proof. Let µ > be fixe. For any small ν >, let us consier the solution provie by Theorem 3.5. We observe that estimate (3.) is inepenent on ν. Hence, by the same argument in the proof of Proposition 3.3, we euce that the family of solutions {ρ µ,ν } ν> is relatively compact in L ( T ). By extracting a convergent a. e. subsequence, one can easily prove the consistency of the limit proceure as ν. Estimate (3.4) is a consequence of Lemma 3.7. Remark 3.. By means of similar arguments as those in Remark 3.9, one can prove that the solution provie by the above theorem enjoys more regularity if so oes the initial atum. Remark 3.. In view of (3.4), the global solution ρ provie in Theorem 3. solves the equation ρ t = (a µ(ρ) ρ ρ( ρ) S), where we have remove the positive part in the term ρ( ρ) Existence via approximation. In this subsection we prove global existence of solutions to the problem (3.) (3.) via approximation by the solution to the regularize problem analyze in the previous subsection. For a fixe n N we efine ρ n := ρ θ n, where θ n is a smooth cutoff function such that θ n (x) if x n, θ n (x) = if x n an θ n otherwise. For a small positive µ, let ρ µ n be the solution to (3.3) provie in the previous subsection with = { x n} an initial atum ρ n. We aim to prove that the family of solutions {ρ µ n} n,µ enjoys the suitable uniform estimates nee to be relatively compact in a certain sense. This time, the estimates on the approximating solution of problem (3.3) evelope in Proposition 3.3 cannot be use to get the esire compactness, since they blow up as µ tens to zero. We aim to improve them in the following Theorem. Theorem 3.3 (Global existence in a close ball). For any fixe integer n, there exists a weak solution ρ n to the problem ρ = (ρ( ρ) (ρ S)) as (x, t) B(, n) [, + ) t S + S = ρ, as (x, t) B(, n) [, + ), (3.5) ρ(x, t) = as x B(, n), ρ(x, ) = ρ (x), with initial atum ρ n. Proof. We prove several energy estimates, obtaine after suitable manipulations of the approximating equation ρ t = (a µ(ρ) ρ ρ( ρ) S). (3.6) In the computations below, as usual, we suppose the solution to be smooth enough in orer to perform integration by parts, thus requiring the initial atum to be

21 CHEMOTAXIS WITH PREVENTION OF OVERCROWDING smooth enough. The result for general initial atum follows by approximation. In what follows, C shall enote a generic positive constant inepenent on µ an on n, an T is a positive time. Step. Multiplying (3.6) by ρ an integrating by parts we obtain ρ x = a µ (ρ) ρ x + C a µ (ρ) ρ x + C S, t ρ( ρ) ρ Sx where we have use the Cauchy Schwarz inequality an the uniform boun for a µ (ρ). Since S H () ρ L (), (3.7) integration over [, T ] an Gronwall inequality yiel T ρ(t) x + a µ (ρ) ρ xt e CT. (3.8) sup t T Step. Given A µ (ρ) = ρ a µ(ξ)ξ, we compute (as in (3.9)) A µ (ρ(t))x + A µ (ρ) x = ρ( ρ) S A µ (ρ)x (3.9) t an, integrating over [, T ] an using (3.8) T T A µ (ρ(t))x + A µ (ρ) x C T A µ (ρ )x + Ce CT. (3.) Step 3. We have the estimate A µ x = A µ (ρ) A µ (ρ) t x = A µ (ρ) t A µ (ρ)x t = ρ t A µ (ρ) t x A µ (ρ) t (ρ( ρ) S) x = a µ (ρ) A µ(ρ) t x A µ (ρ) t ρ( ρ) Sx A µ (ρ) t ( ρ) ρ Sx C A µ (ρ) t x + C ρ x + C ρ( ρ) ρ x, where we have use S L C ρ L, which can be easily obtaine thanks to (.6) an the Green function metho on the ball B(, n) by means of maximum principle for the operator + I, an the uniform boun for ρ. We observe that such an estimate is inepenent on the iameter of. Multiplying by t [, T ] the above computation an integrating over [, T ], using (3.8) an (3.) we get T T ta µ (ρ) t xt C t A µ xt + CT e CT t T C A µ xt + CT e CT C( + T e CT ). (3.) Collecting estimates (3.8), (3.) an (3.), by Sobolev embeing we recover taµ (ρ) t L ( T ).

22 M. BURGER, M. DI FRANCESCO, AND Y. DOLAK Hence, there exists a sequence µ k (as k ) such that A µk (ρ µ k n ) converges almost everywhere in T to some function A(x) as k. Since A µ (ρ) A(ρ) := ρ ρ3 3 as µ, then it is easy to see that A(ρµ k n ) converges to A almost everywhere, an since A has continuous inverse on [, ], this implies that ρ µ k n has an almost everywhere limit efine on T. Such limit is a weak solution of problem (3.5). As usual, it can be easily seen that the solution provie by the above theorem satisfies (3.4). The last step in our approximation argument consists in taking the limit of ρ n as n. We can skip the etails about this proceure, which is rather stanar, an it is base on the same estimate an compactness arguments as above. In fact, it can be easily checke that all the estimates above are uniform with respect to the omain. We are reay to state our final existence theorem. Theorem 3.4 (Global existence of solutions). There exists at least a global weak solution ρ(x, t) to the Cauchy problem (3.) (3.) in the sense of efinition 3.. Proof. We only nee to check that the istributional solution ρ = lim n ρ n satisfies the properties of efinition 3.. Conitions (i) an (ii) can be easily recovere thanks to estimate (3.9), which can be easily obtaine in the limiting case µ =, = R. Conition (iii) is a trivial consequence of the same conition satisfie by the approximating solutions. Conition (iv) is a consequence of (ii). Remark 3.5. The conservation of the total mass can be proven in the same way as in Theorem Entropy solutions. In the following we turn our attention to the problem of uniqueness of suitable solutions. For equations with an interaction of nonlocal fluxes an egenerate iffusions there is no straightforwar way to prove the uniqueness of weak solutions (cf. also [BCM3]) an one might even expect non-uniqueness as for nonlocal transport equations (cf. [DGT]). We therefore turn our attention to a rather natural restriction of weak solutions to so-calle entropy solutions. Apart from uniqueness, the main motivation for consiering entropy solutions is the possibility to obtain correct issipation of entropy functionals, which will be iscusse in the subsections below. Due to the fact that the convolution B ρ is smooth anyway, the behaviour of issipation functionals on this part seems not important an we shall therefore aapt the efinition of entropy solutions for fixe flux S(x, t) (cf. [BCM3] for a more etaile iscussion). Definition 3.6 (Entropy Solutions). We shall say that a nonnegative function ρ L ([, T ] R ) C(, T ; L (R )) is an entropy solution of the Cauchy problem (3.) (3.) on R [, T ] if the following conitions are satisfie: (i) For all c R an all non-negative test functions ϕ Cc ([, T ) R ), the following entropy inequality hols T [ ρ c ϕ t + sign (ρ c) (ρ( ρ) c( c)) S ϕ R ] + ε A(ρ) A(c) ϕ sign (ρ c)c( c) Sϕ x t, (3.) where S = S[ρ] is the unique H solution to S + S = ρ.

23 CHEMOTAXIS WITH PREVENTION OF OVERCROWDING 3 (ii) A(ρ) L (, T ; H (R )) (iii) ρ(x, t) almost everywhere in [, T ] R. (iv) Essentially, as t, R ρ(x, t) ρ (x) x. We start by verifying the existence of entropy solutions. For this sake we take a closer look at our preceing construction of weak solutions. In the first step, namely the nonegenerate approximation on a boune omain, we obtain even a unique classical solution of (3.3) an (3.3). The classical solution clearly satisfies the corresponing entropy conition (with smoothe nonlinearities on the boune omain). Due to a-priori bouns on ρ, A(ρ), an S = B ρ we always extract suitably convergent subsequences when passing from (3.3) an (3.5) an from (3.5) an (3.), so that the entropy inequality (3.) carries over to the limit. Hence, we obtain the following results Theorem 3.7. Let ρ L (R ) satisfy ρ. Then there exists an entropy solution of (3.), (3.) accoring to Definition (3.6). In orer to prove uniqueness of the entropy solution, we shall use continuous epenence of entropy solutions on the flux in the L -norm: Lemma 3.8. Let S, S C(, T ; H (R )) L ([, T ] R ), with S j C(, T ; W, loc (R )) C([, T ] R ), S j L ([, T ] R ) be given an let u j L (, T ; BV (R )) be entropy solutions of u j t = (u j ( u j ) (εu j S j ) ) with initial values u j BV (R ) for j =,. u (t) u (t) L (R ) u u L (R ) + t 4 S S L (,t;bv (R )) where V j = S j L (,t;bv (R )). +t max{v, V } S S L ([,t] R ), (3.3) Proof. The proof can be carrie out in an analogous way to the proof of Theorem.3 in [KR3] by appropriately using the time-epenence of the flux in the estimates an the fact that the function p p( p) has Lipschitz constant an supremum 4 on the interval [, ]. Below we shall prove a uniqueness result in the smaller class of entropy solutions of boune variation in the case of spatial imension one. Before proving their uniqueness, we verify the existence of such entropy solutions: Proposition 3.9 (Regularity of entropy solutions). Let ρ BV (R ; [, ]), then an entropy solution of (3.), (3.) satisfies ρ L (, T ; BV (R )). Proof. We construct the BV-solution by smooth approximation. Let ρ be an L viscosity solution of (3.), (3.) an let ρ δ C(, T ; C (R )) such that ρ δ almost everywhere an ρ δ ρ in C(, T ; BV (R )). Then we also have S δ = B ρ C(, T ; BV (R ))